Thread: finding equivalence classes

1. finding equivalence classes

S is the set of all real #'s

relation R on S is:

R= ((a,b) : a,b are elements of S and a-b is an integer)

R is reflexive, transitive and symmetric, making it an equivalence relation.

how do i find the equivalence classes? i know there are infinitely many.

thanks.

2. Originally Posted by CSkyle
S is the set of all real #'s

relation R on S is:

R= ((a,b) : a,b are elements of S and a-b is an integer)

R is reflexive, transitive and symmetric, making it an equivalence relation.

how do i find the equivalence classes? i know there are infinitely many.

thanks.
Hint: we want a - b = k for some integer k. that means, a = k + b. what do you think forms the equivalence class of the element a?

3. so, the equivalence classes for R consist of all sets (a,b) with a,b being elements of S, that have the form ((k+b),(a-k)) for some integer K ?

4. Originally Posted by CSkyle
so, the equivalence classes for R consist of all sets (a,b) with a,b being elements of S, that have the form ((k+b),(a-k)) for some integer K ?
If $a \in \mathbb{R}, [a] = a + \mathbb{Z}$

For example:
$[\pi] = \{\pi,\pi \pm 1,\pi \pm 2,......\}$